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Carnegie Mellon Center for Nonlinear Analysis
An Entropy Based Theory of the Grain Boundary Character Distribution


Yekaterina Epshteyn
University of Utah
Department of Mathematics

Abstract: Cellular networks are ubiquitous in nature. They exhibit behavior on many different length and time scales and are generally metastable. Most technologically useful materials are polycrystalline microstructures composed of a myriad of small monocrystalline grains separated by grain boundaries. The energetics and connectivity of the grain boundary network plays a crucial role in determining the properties of a material across a wide range of scales. A central problem in materials science is to develop technologies capable of producing an arrangement of grains--a texture--appropriate for a desired set of material properties. Here we discuss the role of energy in texture development, measured by a character distribution. We derive an entropy based theory based on a mass transport and a Kantorovich-Rubinstein-Wasserstein metric to suggest that, to first approximation, this distribution behaves like the solution to a Fökker-Planck Equation.

This is joint work with Katayun Barmak, Eva Eggeling, Maria Emelianenko, David Kinderlehrer, Richard Sharp and Shlomo Ta'asan.